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VO Combinatorics Markus Fulmek VO Combinatorics (Module: “Combinatorics” (MALK)) Markus Fulmek Summer Term 2018 The present (draft of) lecture notes is based on lectures “Combinatorics” held since summer term 2012. It contains selected material from the following text books: Species: Combinatorial Species and tree–like Structures by F. Bergeron, ‚ G. Labelle and P. Leroux [1], Partially ordered sets: Enumerative Combinatorics I by R. Stanley [7], ‚ Asymptotics: Analytic Combinatorics by P. Flajolet and R. Sedgewick ‚ [3]. The reader is assumed to be familiar with basic concepts from Discrete Math- ematics (see, for instance, lecture notes [4]). Moreover, some knowledge from Analysis (power series), Linear Algebra, Algebra (basic group theory) and Com- plex Analysis (contour integrals) is a necessary precondition for the material presented here. This draft was translated to English by Ilse Fischer, Markus Fulmek, Christian Krattenthaler and Michael Schlosser. Many thanks in advance for any hints regarding typos and other flaws: most likely, this draft contains many of them! Vienna, summer term 2019 Contents Chapter 1. Generating functions and Lagrange’s Inversion Formula 1 1.1. A short recap: formal power series 1 1.2. Lagrange’s Inversion formula 1 Chapter 2. Species: Enumeration of combinatorial objects 7 2.1. Motivating Examples 7 2.2. Species, labelled and unlabelled 9 2.3. Unlabelledspeciesandtheenumerationoftrees 10 2.3.1. Enumeration of binary trees 15 2.4. Bijective combinatorics on rooted trees 17 2.4.1. Dyck paths of length 2n 17 2.4.1.1. Planar trees Dyck paths 18 2.4.1.2. Ordered binaryØ trees Planartrees 18 2.5. Unordered rooted trees Ø 18 2.6. labelled species 19 2.6.1. CombinatorialproofofCayley’stheorem 24 2.7. P´olya’s theorem 27 2.8. A generalisation 31 2.9. Cycle index series 32 Chapter 3. Partially ordered sets 43 3.1. Definition and Examples 43 3.2. Construction of posets 48 3.3. Lattices 50 3.3.1. Distributive lattices 54 3.4. IncidencealgebraandM¨obiusinversion 57 3.4.1. Theincidencealgebraoflocallyfiniteposets 58 3.4.2. The zeta function of locally finite posets 59 3.4.3. M¨obius inversion 61 3.4.4. Calculation of M¨obius functions of several concrete posets 63 3.4.5. M¨obius algebra of a locally finite lattice 67 Chapter 4. Asymptotic enumeration 73 4.1. Landau’s notation 73 4.2. Recapitulation:Elementsofcomplexanalysis 76 4.3. Singularity analysis 80 4.3.1. The Exponential Growth Formula 81 4.3.2. Rational functions 83 4.3.3. Meromorphic functions 84 4.3.4. Severaldominantsingularities 89 iii iv CONTENTS 4.3.5. “Standardized”singularityanalysis 89 4.3.5.1. The Gamma–function 90 4.3.5.2. Asymptotics for “standard functions” 93 4.3.5.3. Transfer theorems 100 4.4. Saddle point method 105 4.4.1. Heuristics: contour through saddle point 105 4.4.2. Hayman’s theorem 114 4.4.3. Asaddlepointtheoremforlargepowers 121 4.5. Asymptotics of combinatorial sums 125 4.6. Exp–log scheme 130 Bibliography 133 CHAPTER 1 Generating functions and Lagrange’s Inversion Formula 1.1. A short recap: formal power series Recall that the set C z of formal power series (in some variable z) with com- plex coefficients rr ss C z c zn : c C rr ss “ n ¨ n P #n 0 + ÿě is a commutative algebra with unity over C. The notion “formal” refers to the fact that we do not actually “compute the infinite sum c α ” for some concrete number α, but view c z as a convenient p q p q notation for the sequence of coefficients cn n 0. For some formal power series 2 p q ě c z c0 c1 z c2 z , however, we adopt the intuitive notation c 0 forp q the “ constant` ¨ term` c¨ , i.e.:`¨¨¨c 0 : c . p q 0 p q “ 0 2 THEOREM 1.1.1. The formal power series a z a0 a1 z a2 z possesses a multiplicative inverse if and only if a 0 paq “ 0. ` ¨ ` ¨ `¨¨¨ p q“ 0 ‰ This inverse (if existing) is unique. DEFINITION 1.1.2. Let a z and b z be two formal power series, let b z 0 (i.e., 2 p q p q p q “ b z b1 z b2 z ... , b0 0). Then the composition a b z of a and b is definedp q “ by ¨ ` ¨ ` “ p ˝ qp q a b z : a b z i . p ˝ qp q “ i p p qq i 0 ÿě Note that the condition b 0 0 is necessary, since otherwise we could en- counter infinite sums for thep q coefficients “ of the composition: we do not deal with infinite sums in the calculus of formal power series. THEOREM 1.1.3. Let a z a z a z2 be a formal power series with a 0 p q“ 1 ¨ ` 2 ¨ `¨¨¨ p q“ 0. Then there is a unique compositional inverse b z b z b z2 , i.e. p q“ 1 ¨ ` 2 ¨ `¨¨¨ a b z b a z z, p ˝ qp q“p ˝ qp q“ if and only if a 0. 1 ‰ 1.2. Lagrange’s Inversion formula Let f z be a formal power series with vanishing constant term (i.e., f 0 0): By Theoremp q 1.1.3 we know that there is a (unique) compositional inversep q“F z , i.e., p q F f z f F z z. p p qq “ p p qq “ 1 2 1. GENERATING FUNCTIONS AND LAGRANGE’S INVERSION FORMULA But how can we find this inverse? In fact, there is a quite general and very useful formula which we shall derive in the following: For that, we consider a slightly more general problem, which involves an extension of the calculus of formal power series. DEFINITION 1.2.1. A formal Laurent series (in some variable z) is a (formal) sum a zn n ¨ n N ÿě with coefficients an C, for some N Z: N may be smaller than zero, i.e., there may be negative powersP of z. P We denote the set of all formal Laurent series by C z pp qq Addition, multiplication and composition for formal Laurent series are defined exactly as for formal power series. THEOREM 1.2.2. A formal Laurent series a z possesses a (unique) multiplicative inverse if and only if a z 0. p q p q‰ This implies that the algebra of formal Laurent series is, in fact, a field. In the following we shall always assume that the formal power series f z (with p q f z 0) under consideration “starts with z1, i.e., is of the form p q“ f z f z1 f z2 p q“ 1 ¨ ` 2 ¨ `¨¨¨ f z where f 0 (we might express this as p q 0): The general case 1 ‰ z ‰ m m 1 g z gm z gm 1 z ` for m 1 p q“ ¨ ` ` ¨ `¨¨¨ ą can be easily reduced to this case. 2 LEMMA 1.2.3. Consider some formal power series f z f1 z f2 z where f f 0 0. Then we have for all n Z p q “ ¨ ` ¨ `¨¨¨ 1 “ p q‰ P 1 n 1 z´ f ´ z f 1 z n 0 . r z p q ¨ p q “r “ s ´ ¯ (Here, we made use of Iverson’s notation: “some assertion” equals 1 if “some asser- r s tion” is true, otherwise it equals 01.) n 1 n PROOF. Observe that for n 0 we have n f z f z f z 1: For ‰ ¨ ´ p q ¨ 1 p q “ p p qq n 0, this is a formal power series; and since the coefficient of z 1 equals 0 for ` ˘ ´ everyą formal power series, the assertion is true for all n 0. ą n m 1 1 If n m 0, then f z c m z´ c 1 z´ c0 c1 z ... and “´ ă p q“ ´ ¨ `¨¨¨` ´ ¨ ` ` ¨ ` n m 1 2 f z 1 m c m z´ ´ 1 c 1 z´ c1 ..., p p qq “ p´ q ¨ ´ ¨ `¨¨¨`p´ q ¨ ´ ¨ ` ` so the assertion is also true for all n 0. ă 1Iverson’s notation is a generalization of Kronecker’s delta: δ i j . i,j “r “ s 1.2. LAGRANGE’S INVERSION FORMULA 3 For n 0, we simply compute: “ 1 1 1 f1 2 f2 z z´ f ´ z f 1 z z´ ` ¨ ¨ `¨¨¨ r z p q ¨ p q“ r z f z f z2 1 ¨ ` 2 ¨ `¨¨¨ 1 1 f1 2 f2 z z´ ` ¨ ¨ `¨¨¨ “ r z z f f z 1 ` 2 ¨ `¨¨¨ 1 2 f2 z 1 1 f1 z´ ` ¨ ¨ `¨¨¨ “ r z z f 1 z 2 ´ ´ f1 ´¨¨¨ ˆ ˙ :h z “ p q 1 1 f2 2 z´ 1 loooooooomoooooooon2 z 1 h z h z “ r z z ` f `¨¨¨ ` p q` p q `¨¨¨ ˆ 1 ˙ 1. ´ ¯ “ THEOREM 1.2.4. Let g z be a formal Laurent series and f z f z f z2 p q p q“ 1 ¨ ` 2 ¨ `¨¨¨ be a formal power series with f 0 0, f1 0. Suppose there is an expansion of g z in powers f z , i.e., p q“ ‰ p q p q g z c f k z . (1.1) p q“ k ¨ p q k N ÿě Then the coefficients cn are given by 1 1 n c z´ g1 z f ´ z for n 0. (1.2) n “ nr z p q ¨ p q ‰ An alternative expression for these coefficients is 1 n 1 c z´ g z f 1 z f ´ ´ z for n Z. (1.3) n “ r z p q ¨ p q ¨ p q P PROOF.
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